Statistical Uncertainty in Sampling

• Statistical Uncertainty is the intrinsic variability in measurements or estimates arising from random sampling, noise, or finite data.
• It is quantified in advanced methods like nested sampling using estimators such as Skilling’s and moments-based approaches to assess error bars in Bayesian evidence.
• Accurate uncertainty estimation supports reliable model selection and reproducible results in high-dimensional inference and simulation-driven applications.

Statistical uncertainty refers to the intrinsic, quantifiable variability in the outcome of empirical measurements or computational estimations that arises from the stochastic nature of sampling, noise, or limits imposed by finite data. In advanced research applications, particularly those related to high-dimensional inference, model selection, and simulation-driven domains, statistical uncertainty quantifies the confidence (or lack thereof) in computed outcomes, error bars, or model-derived quantities, and provides a principled basis for error estimation and robustness assessment.

1. Statistical Uncertainty in Nested Sampling

Nested sampling is a Monte Carlo methodology for computing Bayesian evidence (the marginal likelihood), a nontrivial high-dimensional integral central to Bayesian model comparison. The evidence is represented as

$$Z = \int_0^1 L(X)\, dX \approx \sum_i L_i (X_{i-1} - X_i),$$

where $L_i$ are likelihood values and $X_i$ are estimated fractional prior volumes derived from a sequential “peeling” of the prior constrained by increasing likelihood thresholds.

The process introduces stochastic error because the volume contractions at each step are determined by drawing live points subject to the likelihood threshold, with the associated contraction factors $t_j$ independently drawn from $p(t) = M t^{M-1}$, $t \in [0, 1]$ ($M$ being the number of live points).

Two main estimators for the resulting statistical uncertainty in $Z$ are used (Keeton, 2011):

• Skilling’s Estimator: A heuristic, information-theoretic argument gives the fractional uncertainty as $\sigma_Z/Z \approx \sqrt{H/M}$, where $H$ is the Kullback–Leibler information content of the posterior and $M$ is the number of live points.
• Moments-Based Estimator: By computing the mean and variance of $Z$, exploiting the independence of the $t_j$, the new estimator yields

$$\sigma_Z^2 = \frac{2}{M(M+1)} \sum_k L_k (M/(M+1))^k \sum_{i=1}^k L_i ((M+1)/(M+2))^i - \left[\frac{1}{M} \sum_i L_i (M/(M+1))^i\right]^2$$

(Equation Ksig in the original).

Both estimators are computationally inexpensive (no extra simulations required) and, in test cases (Gaussian and log-normal likelihoods), demonstrate strong agreement (e.g., uncertainty estimates of 0.083 and 0.085 for $M=400$).

2. Quantitative Characterization and Error Bar Reporting

Statistical uncertainty in this setting captures fluctuations in the evidence solely from the stochastic nature of the nested sampling procedure. The uncertainty is crucial for:

• Assigning reliable error bars to $Z$ for robust Bayesian model selection.
• Guiding algorithmic choices such as $M$, since the uncertainty scales as $M^{-1/2}$ (increasing $M$ reduces $\sigma_Z / Z$ but requires more computation).
• Ensuring results are reproducible and error estimates are properly propagated in downstream scientific inference.

The propagation of statistical uncertainty follows from the central limit theorem in the large sample limit, but explicit formulas derived from the moments-based estimator provide exact quantification for typical nested sampling runs.

3. Statistical Properties and Derivation

Despite the sequential product structure of $X_i = \prod_{j=1}^i t_j$, the independence of the $t_j$ allows analytic calculation of moments:

• $\langle t^n \rangle = M / (M+n)$,
• $\langle \prod_{j=1}^n t_j \rangle = (M / (M+1))^n$,
• $$\langle \prod_{j=1}^n t_j^2 \rangle = (M / (M+2))^n$$.

By writing

$$Z_k = \sum_i L_i (X_{i-1} - X_i) = \sum_i L_i (\prod_{j=1}^{i-1} t_j - \prod_{j=1}^{i} t_j),$$

the derivation considers expectations over all possible orderings of contraction factors, including covariances between $X_i$ and $X_{i'}$ for $i \neq i'$. This approach accounts for the correlations introduced by the algorithm's sequential structure while leveraging the statistical independence of the underlying draws.

4. Comparative Performance and Test Cases

Rigorous testing on both analytical Gaussian integrals and skewed log-normal likelihoods shows that both the Skilling and moments-based estimators yield nearly identical uncertainty predictions, matching the empirical variance observed over thousands of repeated runs. For example: | Number of live points ($M$) | Info ($H$) | Predicted $\sigma_Z / Z$ (Skilling) | Predicted $\sigma_Z / Z$ (moments) | |-----------------------------|------------|-------------------------------------|------------------------------------| | 400 | 3.6 | 0.094 | 0.094 |

Crucially, fractional uncertainties computed via these estimators are reliable even for non-Gaussian likelihoods, reinforcing the framework’s application to both canonical and complex, heavy-tailed posteriors.

5. Practical Guidelines for Implementation and Application

Both estimators can be computed as a by-product of the standard nested sampling run:

• Skilling's estimator requires calculation of the information $H$ from the weighted set of likelihood samples.
• The moments-based estimator uses the actual sequence of $L_i$ from the run to evaluate the explicit formula for $\sigma_Z^2$.

In practical Bayesian computation:

• Reporting both the point estimate $Z$ and the statistical uncertainty $\sigma_Z$ informs whether differences in evidence between models are significant.
• For applications requiring small error bars (e.g., model selection in cosmology or gravitational wave astronomy), adjusting $M$ is effective, with theoretical guidance provided by the $M^{-1/2}$ scaling.
• Since the moments-based estimator is derived from first principles, it is preferred where rigorous statistical validity is paramount.

6. Broader Implications for Bayesian Inference

Reliable quantification of statistical uncertainty in evidence estimation:

• Strengthens objective model selection by preventing over-interpretation of spurious differences in evidence, especially when error bars overlap.
• Enables principled allocation of computational resources (tradeoff between number of live points, run time, and required accuracy).
• Provides analytic benchmarks by which more sophisticated, possibly parallelized, or adaptive nested sampling algorithms can be assessed.

The agreement between heuristic and moment-based estimators suggests a deep connection between information-theoretic arguments and sampling-based averages. Investigation of formal equivalence between these approaches remains open for future research.

7. Summary Table of Key Results

Aspect	Skilling’s Estimator	Moments-Based Estimator
Formula	$\sigma_Z/Z \approx \sqrt{H/M}$	Explicit sum over $L_i$, $M$
Derivation	Information-theoretic (heuristic)	First-principles analysis
Computational cost	Zero additional	Zero additional
Empirical accuracy	Excellent (tested Gaussian, log-normal)	Excellent
Applicability	General nested sampling	General nested sampling

The rigorous characterization and quantification of statistical uncertainty in nested sampling underpin its utility in advanced Bayesian analysis, model selection, and high-stakes scientific applications requiring explicit error budgets (Keeton, 2011).